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Implementation in CML

First, a selected number, L, say, of weighted maximum likelihood estimations are produced where the weights are Dirichlet random variates. Next, we require an estimate of the covariance matrix of the parameters for the SIR weights. A natural candidate is the maximum likelihood moment matrix (Hartmann and Hartwig, 1995). Portions of this covariance matrix, however, are not available for those parameters whose maximum likelihood estimates are on constraint boundaries. Alternatively, an estimate could be constructed from the average estimate of the covariance matrix from the weighted likelihood re-samples. This is not ideal because, as with the unweighted maximum likelihood covariance matrix of parameters, some of portions of some of the re-sampled covariance matrices will not be available because the re-sampled estimates will be on constraint boundaries.

The implementation in CML computes an estimated covariance matrix directly from the re-sampled estimates. This is not ideal either since some of the estimates will be on constraint boundaries reducing variation. Fortunately, the SIR weights are quite robust in practice to choice of covariance matrix. In any event further research needs to be done on the estimate of the covariance matrix required for the SIR weights.

tex2html_wrap_inline554 weights are calculated for the rows of the weighted bootstrap sample and normalized to sum to one. A Poisson random variate, tex2html_wrap_inline556 for each row is computed with mean tex2html_wrap_inline558 . The final SIR adjusted sample is constructed by defining the frequency of the i-th row of the bootstrap sample as equal to tex2html_wrap_inline560 . Confidence limits and kernel density plots are generated from this sample.

R. Schoenberg
Fri Sep 12 09:47:41 PDT 1997